In this section, we provide a theoretical justification for the effectiveness of GraphArc in enhancing Out-of-Distribution (OOD) detection. Our analysis focuses on two key properties that have been shown to correlate strongly with OOD detection robustness: intra-class variation and inter-class separation (Ming et al., 2022). We demonstrate that GraphArc explicitly minimizes intra-class variation and enlarges inter-class separation through its angular optimization and scaling design.

Proposition 1 (Reduced Intra-Class Variation).
Given normalized feature vectors and angular-based optimization, GraphArc minimizes intra-class variation under a cosine similarity metric.

Proof.
The normalized feature is given by

$$ \tilde{h}_i^{(k)} = \frac{h_i^{(k)}}{||h_i^{(k)}||} $$,
and the normalized class center is

$$ \tilde{W}{y_i} = \frac{W{y_i}}{||W_{y_i}||} $$.

The angular softmax loss is defined as:

$$ \mathcal{L}_{\text{angular}} = \frac{1}{N} \sum_{i} -\log \left( \frac{e^{s \cdot \cos(\theta_{y_i, i})}}{\sum_j e^{s \cdot \cos(\theta_{j, i})}} \right) $$

This loss maximizes $\cos(\theta_{y_i}, i) = \tilde{W}_{y_i}^T \tilde{h}_i^{(k)}$, pulling same-class features toward the same angular direction. Since all class features are drawn toward a shared class anchor on the hypersphere, the angular spread within class $y_i$ is minimized:

$$ \mathbb{E}_{x_i, x_j \sim y_i} \left[ 1 - \cos\left( \angle(\tilde{h}_i^{(k)}, \tilde{h}_j^{(k)}) \right) \right] \to 0 $$

This reduction in intra-class angular variance improves feature compactness and the reliability of confidence calibration under distribution shift.

Proposition 2 (Increased Inter-Class Separation).
Given a Lipschitz continuous feature mapping $\phi$, GraphArc increases the inter-class separation through its use of weight normalization and a scaling factor $s$.

Proof Sketch.
Consider the final layer representation of a GNN:

$$ H^{(L)} = F^{(L)}\left(s \cdot \frac{W^{(L)}}{|W^{(L)}|} H^{(L-1)} Q\right) $$

For two node embeddings \( \phi(x_i), \phi(x_j) \) from different classes, their logits are:

$$ z_i = s \cdot \cos(\theta_{y_i, i}), \quad z_j = s \cdot \cos(\theta_{y_j, j}) $$

Then the angular margin between them becomes:

$$ |z_i - z_j| = s \cdot |\cos(\theta_{y_i, i}) - \cos(\theta_{y_j, j})| $$

As \( s \) increases, the separation in logit space is amplified. Assuming \( \phi \) is Lipschitz continuous with constant \( L \), we have:

$$ |\phi(x_i) - \phi(x_j)| \leq L |x_i - x_j| \quad \Rightarrow \quad |z_i - z_j| \geq s \cdot L \cdot |x_i - x_j| $$

Thus, GraphArc provides a theoretical lower bound on the inter-class margin, proportional to the scaling factor \( s \).

According to Ming et al. (2022), robust OOD detection is facilitated by minimizing intra-class variation \( \mathcal{V}(\phi, \mathcal{E}) \) and maximizing inter-class separation \( \mathcal{I}_\rho(\phi, \mathcal{E}) \). GraphArc satisfies both:

• ✅ Normalization aligns same-class features to a shared direction, minimizing \( \mathcal{V}(\phi, \mathcal{E}) \).
• ✅ Scaling widens angular margins and improves \( \mathcal{I}_\rho(\phi, \mathcal{E}) \).

These properties help GNNs better distinguish OOD samples and generate more calibrated confidence scores.

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